Brauchart–Hardin–Saff conjecture for the minimal Riesz 2-energy on the sphere

Let $\mathcal{E}_2(\mathbb{S}^2,N)$ denote the minimal Riesz $2$-energy of $N$ points on the unit sphere $\mathbb{S}^2$. Let $\gamma$ be the Euler–Mascheroni constant, and let $\gamma_n(a)$ be the generalized Stieltjes constant appearing as the coefficient $\gamma_n(a)/n!$ of $(1-s)^n$ in the Laurent expansion of the Hurwitz zeta function $\zeta(s,a)$ about $s=1$. Define

$$C=\frac{1}{4}\bigl(\gamma-\log(2\sqrt{3}\pi)\bigr)+\frac{\sqrt{3}}{4\pi}\bigl(\gamma_1(2/3)-\gamma_1(1/3)\bigr)=-0.0857\dotso<0.$$

Brauchart–Hardin–Saff conjecture. The following asymptotic expansion holds:

$$\mathcal{E}_2(\mathbb{S}^2,N)=\frac{1}{4}N^2\log N+CN^2+O(1).$$

The conjecture specifies the next-order term after the known leading asymptotic $\mathcal{E}_2(\mathbb{S}^2,N)=\frac14N^2\log N+o(N^2\log N)$. Determining such next-order terms is a central problem in Riesz-energy asymptotics; the stated constant was proposed by Brauchart, Hardin, and Saff, while the general expansion remains unresolved in the source.